The Bateman-type variational formalism for an acoustically-driven drop
By employing the Clebsch potentials, the Bateman-type variational formulation for a drop levitating in an acoustic field is proposed when both fluids, liquid drop and external ullage gas, are barotropic, inviscid, compressible and admit rotational flows.
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Видавничий дім "Академперіодика" НАН України
2023
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irk-123456789-1958622023-12-07T16:15:43Z The Bateman-type variational formalism for an acoustically-driven drop Timokha, A.N. Математика By employing the Clebsch potentials, the Bateman-type variational formulation for a drop levitating in an acoustic field is proposed when both fluids, liquid drop and external ullage gas, are barotropic, inviscid, compressible and admit rotational flows. Використовуючи потенціали Клебша, пропонується варіаційне формулювання типу Бейтмена для краплі, що левітує в акустичному полі, коли обидві рідини, крапля рідини та зовнішній газ є баротропними, нев’язкими, стисливими та допускають вихорові рухи. 2023 Article The Bateman-type variational formalism for an acoustically-driven drop / A.N. Timokha // Доповіді Національної академії наук України. — 2023. — № 3. — С. 17-22. — Бібліогр.: 7 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2023.03.017 http://dspace.nbuv.gov.ua/handle/123456789/195862 532.595 en Доповіді НАН України Видавничий дім "Академперіодика" НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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Математика Математика |
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Математика Математика Timokha, A.N. The Bateman-type variational formalism for an acoustically-driven drop Доповіді НАН України |
| description |
By employing the Clebsch potentials, the Bateman-type variational formulation for a drop levitating in an acoustic
field is proposed when both fluids, liquid drop and external ullage gas, are barotropic, inviscid, compressible and admit
rotational flows. |
| format |
Article |
| author |
Timokha, A.N. |
| author_facet |
Timokha, A.N. |
| author_sort |
Timokha, A.N. |
| title |
The Bateman-type variational formalism for an acoustically-driven drop |
| title_short |
The Bateman-type variational formalism for an acoustically-driven drop |
| title_full |
The Bateman-type variational formalism for an acoustically-driven drop |
| title_fullStr |
The Bateman-type variational formalism for an acoustically-driven drop |
| title_full_unstemmed |
The Bateman-type variational formalism for an acoustically-driven drop |
| title_sort |
bateman-type variational formalism for an acoustically-driven drop |
| publisher |
Видавничий дім "Академперіодика" НАН України |
| publishDate |
2023 |
| topic_facet |
Математика |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/195862 |
| citation_txt |
The Bateman-type variational formalism for an acoustically-driven drop / A.N. Timokha // Доповіді Національної академії наук України. — 2023. — № 3. — С. 17-22. — Бібліогр.: 7 назв. — англ. |
| series |
Доповіді НАН України |
| work_keys_str_mv |
AT timokhaan thebatemantypevariationalformalismforanacousticallydrivendrop AT timokhaan batemantypevariationalformalismforanacousticallydrivendrop |
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2025-07-17T00:07:32Z |
| last_indexed |
2025-07-17T00:07:32Z |
| _version_ |
1837850537335193600 |
| fulltext |
17ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2023. № 3: 17—22
C i t a t i o n: Timokha A.N. Th e Bateman-type variational formalism for an acoustically-driven drop. Dopov. Nac.
akad. nauk Ukr. 2023. № 3. С. 17—22. https://doi.org/10.15407/dopovidi2023.03.017
© Publisher PH «Akademperiodyka» of the NAS of Ukraine, 2022. Th is is an open access article under the CC BY-
NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)
МАТЕМАТИКА
MATHEMATICS
https://doi.org/10.15407/dopovidi2023.03.017
UDC 532.595
A.N. Timokha, https://orcid.org/0000-0002-6750-4727
Institute of Mathematics of the NAS of Ukraine, Kyiv, Ukraine
E-mails: tim@imath.kiev.ua, atimokha@gmail.com
Th e Bateman-type variational formalism
for an acoustically-driven drop
By employing the Clebsch potentials, the Bateman-type variational formulation for a drop levitating in an acoustic
fi eld is proposed when both fl uids, liquid drop and external ullage gas, are barotropic, inviscid, compressible and admit
rotational fl ows.
Keywords: Bateman variational principle, Clebsch potentials, acoustical levitation.
Idea of the present paper comes from [1, 2] whose objects are two oscillating compressible ideal baro-
tropic fl uids when an acoustic vibrator is located in one of them to both generate a high-frequency
acoustic fi eld and govern the interface motions. Physically, these two papers deal with an acoustical
positioning of a large liquid mass (volume) in microgravity conditions and an acoustically-levitating
drop, respectively. Irrotational fl uid fl ows are assumed that made it possible to show how to de-
rive the corresponding free-interface boundary value problem based on hydrodynamic variational
principles of Hamilton-Ostrogradskii’ and Bateman’s types. Specifi cally, the Hamilton-Ostrogradskii
principle requires a kinematic constraint but the Bateman’s ones derive the complete free-interface
boundary value problem. Th e latter fact makes the Bateman-type principles of especial interest as
being important for the multimodal modelling in the liquid sloshing dynamics and, through separa-
tion of fast and slow times directly in the action, for deriving a quasi-potential energy functional of
the so-called vibro-equilibria, which are time-averaged interfaces between the two fl uids that diff er
in the considered cases from capillary interface shapes governed by gravitation and surface tension.
Assuming rotational fl ows for acoustically-levitating drops can be important due vortices in
fl uids and/or rotation of the liquid drop itself [3, 4]. Th is assumption requires a generalization of
the Bateman-type variational principles like it has recently been done in [5] for the liquid sloshing
problem. Such a generalization is proposed in the present paper.
Th roughout the forthcoming text, two compressible barotropic fl uids with, possibly, rotation-
al fl ows, external ullage gas 1 ( )Q t and liquid drop 2 ( )Q t , are considered in the inertial Oxyz co-
18 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2023. No 3
Timokha A.N.
ordinate frame as illustrated in Fig. 1. Here, Σ(t) denotes the unknown a priori interface between
fl uids defi ned implicitly by the equality Z ( , , , ) 0x y z t and the prescribed surface ( )S t bounding
the ullage gas whose vibrational motions are described by the equality ( , , , ) 0Y z y z t , where Y is
the prescribed function. Th e outer normal vectors n are determined by / | | on Σ(t) and
/ | |Y Y on ( )S t , respectively.
Th e two fl uids are compressible with densities 1 ( , , )x y z and 2 ( , , )x y z , so that the mass
conservation
( )
1, 2
i
i i
Q t
dQ M i (1)
can be treated as geometric constraints; ( , , ) – , ( , , )U x y z x y z g r r is responsible for the grav-
ity fi eld.
Th e velocity fi elds in ( )iQ t are (non-uniquely [3]) governed by the Clebsch potentials
, , , , ( , , , )( )i ix y z t m x y z t , and , , , , 1, 2( )i x y z t i as follows
i i i im v . (2)
Based on [6, p. 47], the following Bateman-type Lagrangian is introduced
2
2
1 ( )
[( ( )1, , , , ) – ,
2
]
i
i i i i i t i i t i i i
i Q t
L m Z m U E dQ
iv (3)
where ( )i iE is the inner energy of the barotropic fl uids for which the pressure is postulated by
2 ' ( )i i i ip E . (4)
Th e Lagrangian (3) yields the action
2
1
2
1
( , , , , ) [ ( , , , , ) ]
t
i i i i i i i i i i
t i
W m L m Z M dt
for 1 2 ,t t (5)
where ( )i i t are the Lagrange multipliers caused by
the geometric constraints (1). Th e action (5) is a func-
tion of the fl uid densities, the Clebsch potentials and the
instant free-interface shape. Th e zero fi rst variation of
the action (5) by , , ,i i i im , and should derive the
free-interface boundary value problem, which describes
behaviour of the two fl uids due to prescribed vibrational
motions of ( )S t .
Remark 1. In contrast to the Bateman-type variational
formulation for irrotational fl ows [1], the acoustical vi-
brator cannot be determined via the Neuman boundary
condition on a fi xed gas box surface with anppropriate in-
tegral in the Lagrangian (3). One should instead introduce
the moving surface ( )S t .
S(t)n
n
Σ(t)
Q1(t)
Q2(t)
Fig. 1. Schematic sketch of a levitating drop
and introduced notations
19ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2023. № 3
Th e Bateman-type variational formalism for an acoustically-driven drop
Henceforth, we assume that the Clebsch potentials are smooth functions in 1 2( ) ( )Q t Q t .
Th is implies in particular that these functions can be analytically continued through the smooth
interface ∑( )t . Using the calculus of variables, specifi cally, the Reynolds transport and divergence
theorems [7, Appendix A], makes it possible to establish the following propositions.
Lemma 1. Under the smoothness assumption above, the zero fi rst variation condition
1 2,
0 0
i i t t
W subject to (6)
is equivalent to the continuity equation
( ) 0 ( )t i i i iin Q t v , (7)
and the normal-velocity conditions
1 ( ) ( ); ( ) ( ). t t
i
Yon t a on S t b
Y
v n v n (8)
Th e proof is based on the following derivation line with substituting the second condition of (6):
2
1 (
2
1 )
[ ( )]
i
t
i t i i i
i Q tt
dQdt
v
2
1
2
( ) ( )1
[ ]( )
i i
t
i i i t i i i
i Q t Q tt
d dQ dQ
dt
v
1 1 1
Ó( ) ( )
Æ Y( 1)
Æ Y
i t t
i i i
t S t
n dS n dS
v v dt.
Lemma 2. Under the smoothness assumption above, the zero fi rst variation condition
0
im W (9)
is equivalent to the equations
0t i t i i id v , (10)
which implies that the Clebsch potentials i remain constant values during motions of liquid parti-
cles (the vortex lines move with fl uids and always contain the same particles).
Th e proof is based on the expression for this fi rst variation
2
1 )
2
1 (
[ ] 0.
i
t
i i t i i i
i Q tt
m dQ dt
v
Z
Z∑
20 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2023. No 3
Timokha A.N.
Lemma 3. Under the smoothness assumption above and the zero variational condition (6) for
the action [equivalent to (7) and (8)], the zero fi rst variation condition
1 2,
0 0
i i t t
W subject to (11)
is equivalent to
0t i t i i id m m m v , (12)
which has the same hydrodynamic meaning like (10) but for the Clebsch potential im .
Th e proof uses is the following derivation line
2
1
2
1 (
2
1
2
1
1 1
( )
) (
1
)
)
1
(
[ ( )]
[ ( )]
| |
( )
i
i i
t
i i t i i i i
i Q tt
t
i i i i t i i i i i
i Q t Q tt
t
S t
m m dQdt
d m dQ m m dQ
dt
Ym n dSdt
Y
v
v
v
Ó( )
( 1)i t
i i i i
t
m n dS
v
together with the second condition of (11) and (8) to show that
( ) [ ( )] [ ] 0 ( )t i i i i i i t i i i i t i i im m m m m v v v
that, accounting for (7), deduces (12).
Lemma 4. Under the smoothness assumption above, the zero fi rst variation condition
0
i
W (13)
is equivalent to the equality
2 ' 0( ) ( ) (
2
)1
t i i t i i i i i i im U E E t iv in ( )iQ t , (14)
which can be treated as the Bernoulli equation (Lagrange-Cauchy integral) of the Euler equation
( )i
t i t i i i i
i
pd U in Q t
v v v v (15)
provided by (9), (11) and defi nition (4).
Th e proof of (14) becomes obvious aft er taking the variation of (5) by i . To prove (15), one
should apply the gradient operation to equality (14) and defi nition (4). Th e second application
yields the derivation line
' ''( ) ( )[2 ]i
i i i i i i
i
p E E
[ '( )) (i i i i iE E ].
∑
21ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2023. № 3
Th e Bateman-type variational formalism for an acoustically-driven drop
Furthermore, the left -hand side of (15) can be re-written as follows
( )
[ ] { ( )) }(
i i i i i
i i i i t i i t i t i i i i i i
m m
d d m m m m
i i iv v v
v v
[ ]t i i t i i i i i i i im m dm v v (16)
and, applying the gradient operation to the fi rst three quantities in (14) gives
21 [ ]
2t i i t i t i i t i t i i im m m
i iv v
) . ( [ ]i i i i i t i i t i i i i i im m m m m d i i iv v v v (17)
Th e right-hand sides in (16) and (17) are identical provided by (10), (12) following from the
zero-variation conditions (9) and (11).
Lemma 5. Under the smoothness assumption above, the zero fi rst variation condition
0W (18)
is equivalent to the interface condition
2
1 1 1 1 1 1 1 1[ ( ) ( )]1
2t tm U E t v
2
2 2 2 2 2 2 2 2 ]1 ([ ) )(
2t tm U E t v on ( )t , (19)
which is the same as the traditional dynamic interface condition
1 2p p on ( )t (20)
provided by (6), (9), (11) and defi nition (5).
Proof. Equality (19) obviously follows from the zero-variation condition (18) by :
2
1
2
2
1 ( )
1 (–1)( ) 0. )
2
[
|
(
|
]
i
t i
i t i i t i i i i
it Q t
m U E t dQ
iv
In order to deduce (20), one should note that defi nition (5) and Bernoulli equation for barotropic
compressible fl uids (14) derive
21[ ] –
2
( ) ( )i t i i t i i i i i im U E t p v
provided, according to conditions of the Lemma 4, by (6), (9), (11).
Summarizing the Lemmas 1-5 shows that the Bateman-type variational formulation derives
a free-interface boundary value problem on a drop oscillating in an acoustic fi eld excited by pre-
22 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2023. No 3
Timokha A.N.
scribed vibrations of the box surface S(t) as shown in Fig. 1 by consequently applying the neces-
sary condition (6), (9), (11), (13), and (16) to the action (5). Th e main result can be formulated as
the following theorem.
Th eorem 6. Under the smoothness assumption above, the zero fi rst variation of the action (5),
0
i i i im ZW W W W W W subject to
1 2,
0i t t
and
1 2,
0 ,i t t
is equivalent to the free-interface boundary value problem on acoustically-driven liquid drop 2 ( )Q t in ul-
lage gas 1 ( )Q t for a prescribed vibration of the gas box on ( )S t . Th e diff erential boundary value problem
consists of the continuity equations (7) in fl uid domains, the kinematic boundary condition (8a) on the
interface and the `vibrating box surface’ condition (8b), the Bernoulli equations (14) (alternatively, the
Euler equations (15)) in fl uid domains, the dynamic interface condition (20) on the interface as well as
the vortex line conditions (10) and (12) provided by the defi nitions of pressure (5) and velocity fi elds (2).
Conclusions and discussion. By using the Clebsch potentials, the Bateman-type variational
formulation from [1, 2] can be generalised for barotropic fl uids to the case of rotational fl uid fl ows.
Th e free-interface boundary value problem derived from the Bateman-type variational formula-
tion not necessary has a unique solution. One should then consider viscous fl uid fl ows.
Th e author acknowledges the fi nancial support of the National Research Foundation of Ukraine
(Project number 2020.02/0089).
REFERENCES
1. Lukovskii, I. A. & Timokha, A. N. (1993). Variational formulations of nonlinear boundary-value problems with
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1849-1860. https://doi.org/10.1007/BF01061355
2. Chernova, M. O, Lukovsky, I. A. & Timokha, A. N. (2015). Differential and variational formalism for acoustical-
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3. Pandey, K., Prabhakaran, D. & Basu, S. (2019). Review of transport processes and particle self-assembly in acous-
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4. Chen, H., Li, A., Zhang, Y. & Xiaoqiang, Z. (2022). Evaporation and liquid-phase separation of ethanol-cyclohex-
ane binary drops under acoustic levitation. Phys. Fluids, 34, Iss. 9, art. 092108. https://doi.org/10.1063/5.0109520
5. Timokha, A. N. (2022). A note on the variational formalism for sloshing with rotational flows in a rigid tank with
unprescribed motion. Ukr. Math. J., 73, No. 10, pp. 1580-1589. https://doi.org/10.1007/s11253-022-02015-3
6. Bateman, H. (1944). Partial differential equations of mathematical physics. New York: Dover Publications, 556 pp.
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Received 17.04.2023
О.М. Тимоха, https://orcid.org/0000-0002-6750-4727
Інститут математики НАН України, Київ
E-mails: tim@imath.kiev.ua, atimokha@gmail.com
ВАРІАЦІЙНИЙ ФОРМАЛІЗМ ТИПУ БЕЙТМЕНА ДЛЯ АКУСТИЧНО КЕРОВАНОЇ КРАПЛІ
Використовуючи потенціали Клебша, пропонується варіаційне формулювання типу Бейтмена для краплі,
що левітує в акустичному полі, коли обидві рідини, крапля рідини та зовнішній газ є баротропними,
нев’язкими, стисливими та допускають вихорові рухи.
Ключові слова: віаріаційний принцип Бейтмена, потенціали Клебша, акустична левітація.
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